Re: logging in

As for the original statement, Qwerty is just right...that always works for me
Based on what I know of the minecraft code:

Bad login means the server is getting confused about who you are and your identity - therefore restarting minecraft will assure the system of your identity
End of stream sometimes happens when, i believe, the minecraft applet is attempting to regulate the amount of times you try and fail to connect to the server, because this can cause a server crash if too many people do this. Either that or end of stream means literally end of stream; the stream of data from the server you were trying to connect to has ended.

Most of the time it is not necessary to restart the router; I have tested this by using my phone (which works as a router) and when i get that error message i switch from my home router to my phone, and It always says the same error message.

Re: logging in

I have needed to restart my router before so it sometimes is the issue. As you said the end of stream is the stream of data from the server to your computer. One of the places it goes to get to your computer is through your router so that is sometimes where the issue lies